March 21, 2022

Disperal

  • “… the movement an animal makes from its point of origin to the place where it reproduces or would have reproduced had it survived and found a mate.” – Howard 1960

Role of dispersal …

  • Colonizers: pioneer new habitat patches as favorable patches wink out

  • Demographic buffers: reduce impact of mortality in prime-age individuals by providing a pool of ready replacements

  • Rescuers: provide new breeders or new genetics to isolated sub-populations that are declining.

Without disperal all metapopulations will eventually go extinct.

Two (?) types of disperser

  • Environmental Dispersers: disperse when environmental conditions deteriorate
  • Innate Dispersers: “hard-wired” to disperse

Functional genomics

Alleles of Pgi gene in butterfly trade off metabolic rate (low = better disperser) with clutch size (Haag et al. 2005). Also, linked to probability of colonizing and heritability.

Typical features of dispersing individuals

  • Sub-adult
  • High mortality
  • Straddling brave and foolish

See: the loneliest California sea lion in the world (Zalophus californianus)

Unpacking boldness

Bolder individuals MORE likely to disperse

aggressive / less neophobic

  • Rhesus macaques, Macaca mulatta (Mehlman et al. 1995)
  • Mountain bluebirds, Sialia currucoides (Duckworth & Badyaev, 2007)
  • Killifish, (Skalski 2001)
  • Great tit, Parus major, (Dingemanse et al., 2003)
  • Roe deer, Capreolus capreolus (Debeffe et al. 2014)

But, agressive house mice Mus musculus domesticus drive away more timid mice (Pocock, Hauffe, & Searle, 2005)

Facial expressions of mice in aggressive and fearful contexts (Defensor 2012)

Is boldness only “innate”?

Plants certianly also disperse!

  • Allochory (external):
    • anemochory: Dispersal by wind (dandelions)
    • hydrochory: Dispersal by water (coconuts)
    • zoochory: Dispersal by animal (poor avocados!)
  • Autochory (self-powered:
    • Dispersal by explosion - (peas in pods)
    • Dispersal by gravity: falling fruits break and disperse seeds

Invasion of the chories

Autochory (self dispersal)

  • Barochory (transport via gravity) - Blastochory (dispersal via runners) - Herpochory (transport via active creeping)

Ballochory (self seeder)

  • Zooballochory (impetus provided by animals) - Anemoballochory (impetus provided by wind) - Hydroballochory (impetus provided by water) - Autoballochory (propulsion mechanisms based on sap pressure or drying)

  • Ethelochory (deliberate dispersal) - Speirochory (dispersal due to accidental dispersal with seeds) - Agochory (unintentional dispersal)

Allochory (dispersal via a vector)

  • Anemochory (transport via wind) - Chamaechory (transport close to the soil because of large size and soil adherence to animals, etc.) - Meteochory (transport in air for small seeds) - Boleochory (transport started by wind, further dispersal assured by other mechanisms) - Hydrochory (transport in water) - Nautochory (transport by movement in the sea) - Bythisochory (drifting in flowing water) - Ombrochory (transport via rain drops)

Zoochory (dispersal via animals)

  • Ornithochory, myrmecochory, etc. - Epichory (transport by attachment of propagules to animals) - Endochory (transport of propagules during passage through the gut) - Stomatochory (transport in the mouth) - Dysochory (transport of accidentally ingested propagules) - Hemerochory (dispersal by man)

Modeling dispersal: the random walk

The solution - Diffusion

A slowly widening bell-shaped curve

1D diffusion

2D diffusion

Drunkard’s Walk

Also known as “Brownian Motion”

The 1-D probability of being in location \(x\) after time \(t\) given a random walk with diffusion constant \(D\) is:

Diffusion applied to invasion speed

Diffusion + colonization: \(\sqrt{Area} \propto t\)

But diffusion assumes all individuals are identical

Homogeneous Random Walk ...

In fact, all individuals are unique

Heterogeneous Random Walk ...

Heterogeneity leads to fat tails …

Not \(Pr(x) \sim \exp(-x^2)\) but closer to \(Pr(x) \sim \exp(-x^\kappa)\) where \(0 < \kappa < 2\).

… and fatter tails lead to faster invasions

A fat-tailed model can actually lead to an accelerating invasion front!

Accelerating Cane Toad Invasion!

(Skip to 8:45)

Shift topics …. to Disease

SIR model

Everyone in a population of fixed size \(N\) is either:

  • Susceptible, who (can become)
  • Infected after which they (can become)
  • Removed.

(Removed are either Recovered or Recently Deceased)

\[ \left\{\begin{aligned} & \frac{dS}{dt} = - \frac{\beta I S}{N}, \\[6pt] & \frac{dI}{dt} = \frac{\beta I S}{N}- \gamma I, \\[6pt] & \frac{dR}{dt} = \gamma I, \end{aligned}\right. \]

Note - the similarity (simplification) relative to predator-prey models. How many parameters are there?

Note - the sum of the three rates \({dS \over dt} + {dI \over dt} + {dR\over df} = 0\), because population is fixed at \(S+I+R = N\) (none of the crazy fluctuations of Lotka-Volterra).

SIR Model Graphical

Eventually EVERYONE will be Removed! (good news? bad news?)

General families of Compartmental Models

And with that ….